Initial Setup & Theoretical Foundation

We are required to find exactly six rational numbers strictly between the integers $3$ and $4$.

[Per the Density Property of Rational Numbers], between any two distinct rational numbers, there exists an infinite number of rational numbers. An integer can always be expressed as a rational number $\frac{p}{q}$ where $q = 1$. Therefore, we can write the given numbers as:

• $a = \frac{3}{1}$
• $b = \frac{4}{1}$

Step 1: Applying the Common Denominator Algorithm

To find $n$ evenly spaced rational numbers between two integers $a$ and $b$, a highly reliable algebraic method is to convert both integers into equivalent fractions sharing a common denominator of $n + 1$.

Given that we need $n = 6$ rational numbers, we calculate our target multiplier:

$n + 1 = 6 + 1 = 7$

We now multiply both the numerator and the denominator of our initial rational expressions by $7$ to maintain equivalence [By the Multiplicative Identity Property, multiplying by $\frac{7}{7}$ is equivalent to multiplying by $1$]:

$3 = \frac{3 \times 7}{1 \times 7} = \frac{21}{7}$

$4 = \frac{4 \times 7}{1 \times 7} = \frac{28}{7}$

Step 2: Identifying the Intermediate Rational Numbers

We have successfully established the boundary values as $\frac{21}{7}$ and $\frac{28}{7}$. Because the denominators are identical, any fraction with a numerator strictly between $21$ and $28$ (over the denominator $7$) will be a rational number lying strictly between $3$ and $4$.

Incrementing the numerator by $1$ sequentially yields exactly six rational numbers:

$\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7}$

Visual Representation: The Rational Number Line

The following scale demonstrates the exact geometric placement of these rational numbers, dividing the segment between $3$ and $4$ into $7$ equal sub-intervals.

Alternative Method (Decimal Expansion)

While the fractional method is the standard algebraic approach, rational numbers can also be expressed as terminating decimals. Any terminating decimal between $3.0$ and $4.0$ is a valid rational number. For example:

$3.1 = \frac{31}{10}, \quad 3.2 = \frac{32}{10} = \frac{16}{5}, \quad 3.3 = \frac{33}{10}, \quad 3.4 = \frac{34}{10} = \frac{17}{5}, \quad 3.5 = \frac{35}{10} = \frac{7}{2}, \quad 3.6 = \frac{36}{10} = \frac{18}{5}$

Both sets of numbers are mathematically correct due to the infinite density of rational numbers.

Final Solution: Six rational numbers between $3$ and $4$ are $\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \text{ and } \frac{27}{7}$.

Step 1: Formal Definitions of the Number Sets

To evaluate the validity of the statement, we must first establish the rigorous mathematical definitions of the two sets of numbers in question: Integers and Whole Numbers.

• Integers ($\mathbb{Z}$): The set of integers includes all positive counting numbers, their negative counterparts, and zero. It extends infinitely in both positive and negative directions.
  $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$
• Whole Numbers ($\mathbb{W}$): The set of whole numbers includes all natural (counting) numbers and zero. It does not contain any negative numbers.
  $\mathbb{W} = \{0, 1, 2, 3, \dots\}$

Step 2: Set-Theoretic Analysis

The statement "Every integer is a whole number" can be translated into set theory as a subset proposition: $\mathbb{Z} \subseteq \mathbb{W}$ [Read: The set of Integers is a subset of the set of Whole Numbers].

For $\mathbb{Z} \subseteq \mathbb{W}$ to be true, every element $x$ that belongs to $\mathbb{Z}$ must also belong to $\mathbb{W}$ ($\forall x \in \mathbb{Z} \implies x \in \mathbb{W}$).

By comparing the sets defined in Step 1, we observe the following relationship:

• All elements of $\mathbb{W}$ exist within $\mathbb{Z}$. Therefore, $\mathbb{W} \subset \mathbb{Z}$ is a true statement [Every whole number is an integer].
• However, $\mathbb{Z}$ contains negative numbers (e.g., $-1, -2, -3$) which do not exist in $\mathbb{W}$. Therefore, $\mathbb{Z} \not\subseteq \mathbb{W}$.

Step 3: Visual Proof via Number Line

The following high-precision number line illustrates the spatial and logical boundaries of both sets. Notice that the domain of Integers extends into the negative axis, whereas Whole Numbers are strictly bounded at zero and extend only in the positive direction.

Step 4: Counterexample Formulation

In formal logic, a universal affirmative statement ("Every A is B") is proven false if even a single counterexample can be provided. We must find an integer that is not a whole number.

Let $x = -5$.

• Is $-5 \in \mathbb{Z}$? Yes, $-5$ is a negative integer.
• Is $-5 \in \mathbb{W}$? No, the set of whole numbers $\mathbb{W}$ does not contain negative values.

Because we have identified an element that belongs to $\mathbb{Z}$ but not to $\mathbb{W}$, the statement "Every integer is a whole number" is logically invalid.

Final Solution: False. The statement is false because the set of integers includes negative numbers (e.g., $-1, -2, -3$), whereas the set of whole numbers only includes zero and positive counting numbers. Therefore, negative integers are not whole numbers.

Step 1: Define the Mathematical Sets

To evaluate the truth value of the statement, we must first establish the rigorous mathematical definitions of the sets involved.

• Natural Numbers ($\mathbb{N}$): This is the set of all counting numbers starting from $1$ and extending to infinity.
  Mathematically, $\mathbb{N} = \{1, 2, 3, 4, \dots\}$
• Whole Numbers ($\mathbb{W}$): This is the set of all natural numbers along with the number zero ($0$).
  Mathematically, $\mathbb{W} = \{0, 1, 2, 3, 4, \dots\}$

Step 2: Analyze the Set Relationship

By comparing the elements of both sets, we can observe the structural relationship between them.

The set of whole numbers can be expressed as the union of the set containing zero and the set of natural numbers:
$\mathbb{W} = \{0\} \cup \mathbb{N}$

[By the definition of subsets in set theory], if every element of set $A$ is also an element of set $B$, then $A$ is a subset of $B$ ($A \subseteq B$). Since every element in $\mathbb{N}$ ($1, 2, 3, \dots$) is explicitly contained within $\mathbb{W}$, we can state that $\mathbb{N}$ is a proper subset of $\mathbb{W}$ ($\mathbb{N} \subset \mathbb{W}$).

Step 3: Visual Representation via Euler-Venn Diagram

The following diagram illustrates the subset relationship, demonstrating that the boundary of Natural Numbers is entirely enclosed within the boundary of Whole Numbers.

Step 4: Formulate the Logical Conclusion

Because the collection of whole numbers contains all the natural numbers in addition to zero, there is no natural number that is not also a whole number. Therefore, the premise that "every natural number is a whole number" is logically sound and mathematically verified.

Final Solution: True. The statement is true because the set of whole numbers consists of all natural numbers and the number zero ($\mathbb{W} = \mathbb{N} \cup \{0\}$). Therefore, every natural number inherently belongs to the set of whole numbers.

Given Variables & Initial Setup

We are tasked with finding exactly five rational numbers that lie strictly between the following two given rational numbers:

• Lower bound: $a = \frac{3}{5}$
• Upper bound: $b = \frac{4}{5}$
• Required number of rational numbers: $n = 5$

[Per the Density Property of Rational Numbers, between any two distinct rational numbers, there exist infinitely many rational numbers. To find a specific finite set of rational numbers spaced evenly between two fractions with identical denominators, we utilize the $(n + 1)$ scaling method.]

Step 1: Determining the Scaling Factor

To find $n$ rational numbers between two fractions that already share a common denominator, we must amplify the fractions by multiplying both the numerator and the denominator by a scaling factor of $(n + 1)$.

Given $n = 5$:

$\text{Scaling Factor} = n + 1 = 5 + 1 = 6$

Step 2: Creating Equivalent Fractions

We apply the scaling factor to both the lower and upper bounds to generate equivalent fractions with a larger common denominator. This expands the "gap" between the numerators, allowing us to identify integers between them.

For the lower bound $a$:

$a = \frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}$

For the upper bound $b$:

$b = \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30}$

[By the Fundamental Property of Fractions, multiplying the numerator and denominator by the same non-zero integer preserves the value of the rational number.]

Step 3: Identifying the Intermediate Rational Numbers

We now look for five rational numbers strictly between $\frac{18}{30}$ and $\frac{24}{30}$. Since the denominators are identical, we simply increment the numerator by $1$ starting from $18$ up to $23$:

$\frac{18}{30} < \frac{19}{30} < \frac{20}{30} < \frac{21}{30} < \frac{22}{30} < \frac{23}{30} < \frac{24}{30}$

Thus, the five intermediate rational numbers are:

$\frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30}, \frac{23}{30}$

Step 4: Simplification to Lowest Terms (Optional but Recommended)

To adhere to standard mathematical conventions, we reduce each intermediate fraction to its simplest form by dividing the numerator and denominator by their Greatest Common Divisor (GCD).

• $\frac{19}{30}$ (Already in simplest form, as 19 is prime)
• $\frac{20}{30} = \frac{20 \div 10}{30 \div 10} = \frac{2}{3}$
• $\frac{21}{30} = \frac{21 \div 3}{30 \div 3} = \frac{7}{10}$
• $\frac{22}{30} = \frac{22 \div 2}{30 \div 2} = \frac{11}{15}$
• $\frac{23}{30}$ (Already in simplest form, as 23 is prime)

Visual Representation: Number Line

The following high-precision SVG illustrates the exact spatial distribution of these rational numbers on a number line.

Final Solution: The five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$ are $\frac{19}{30}$, $\frac{20}{30}$, $\frac{21}{30}$, $\frac{22}{30}$, and $\frac{23}{30}$. (Expressed in simplest form: $\frac{19}{30}$, $\frac{2}{3}$, $\frac{7}{10}$, $\frac{11}{15}$, and $\frac{23}{30}$).

Step 1: Theoretical Foundation of Rational Numbers

In mathematics, the set of rational numbers, denoted by $\mathbb{Q}$, is defined as the set of all numbers that can be expressed as a quotient or fraction $\frac{p}{q}$ of two integers. [Per the fundamental axioms of Number Theory], the formal definition requires two strict conditions to be met:

• The numerator $p$ and the denominator $q$ must be integers ($p, q \in \mathbb{Z}$).
• The denominator $q$ must not be equal to zero ($q \neq 0$), because division by zero is mathematically undefined.

Step 2: Analyzing Zero ($0$) Against the Rational Criteria

To determine if zero is a rational number, we must test it against the two conditions established in Step 1.

• Condition 1 (Integer Numerator): We can set the numerator $p = 0$. Since zero is a whole number without a fractional or decimal component, it is an integer ($0 \in \mathbb{Z}$).
• Condition 2 (Non-Zero Integer Denominator): We can select any non-zero integer for the denominator $q$ (e.g., $1, 2, -5, 100$). For any chosen integer where $q \neq 0$, the second condition is perfectly satisfied.

Step 3: Constructing the $\frac{p}{q}$ Form

When zero is divided by any non-zero number, the quotient is always zero. [By the Zero Property of Division], we can express $0$ in infinitely many equivalent fractional forms:

$0 = \frac{0}{1} = \frac{0}{2} = \frac{0}{-7} = \frac{0}{999}$

In every single one of these representations, $p = 0$ and $q \in \{1, 2, -7, 999\}$. Because $p$ and $q$ are integers and $q \neq 0$, zero rigorously fulfills the definition of a rational number.

Step 4: Visual Representation on the Number Line

The number line below illustrates how zero sits among the integers, while simultaneously being representable as a rational fraction.

Final Solution: Yes, zero is a rational number. It can be written in the form $\frac{p}{q}$ (such as $\frac{0}{1}$, $\frac{0}{2}$, or $\frac{0}{-5}$), where $p = 0$ and $q$ is any non-zero integer ($q \neq 0$).

Step 1: Formal Definitions of the Number Sets

To evaluate the validity of the statement, we must first establish the rigorous mathematical definitions of the sets involved:

• Rational Numbers ($\mathbb{Q}$): A rational number is any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers ($\mathbb{Z}$) and $q \neq 0$. Examples include $\frac{1}{2}$, $-\frac{3}{4}$, $5$ (since $5 = \frac{5}{1}$), and $0$.
• Whole Numbers ($\mathbb{W}$): The set of whole numbers consists of all non-negative integers. It is defined as $\mathbb{W} = \{0, 1, 2, 3, \dots\}$.

Step 2: Logical Analysis of the Proposition

The given statement asserts that "Every rational number is a whole number." In set theory notation, this proposition claims that the set of rational numbers is a subset of the set of whole numbers ($\mathbb{Q} \subseteq \mathbb{W}$).

[Per the axioms of set theory], for $\mathbb{Q} \subseteq \mathbb{W}$ to be true, every single element that exists in $\mathbb{Q}$ must also exist in $\mathbb{W}$. If we can find even one element in $\mathbb{Q}$ that is not in $\mathbb{W}$, the statement is proven false by counterexample.

Step 3: Disproving via Counterexamples

Let us test specific rational numbers against the definition of whole numbers to find a counterexample:

• Fractional Values: Consider the number $\frac{3}{5}$. It is a rational number because it is in the form $\frac{p}{q}$ where $p=3$, $q=5$, and $q \neq 0$. However, $\frac{3}{5} = 0.6$, which is a decimal value and not a non-negative integer. Therefore, $\frac{3}{5} \notin \mathbb{W}$.
• Negative Integers: Consider the number $-4$. It is a rational number because it can be written as $\frac{-4}{1}$. However, whole numbers strictly include only non-negative integers. Therefore, $-4 \notin \mathbb{W}$.

Step 4: Visualizing the Set Hierarchy

The relationship between these number systems is strictly hierarchical in the opposite direction: $\mathbb{W} \subset \mathbb{Q}$. Every whole number is a rational number, but the converse is not true. The Euler diagram below illustrates this structural relationship, proving that $\mathbb{Q}$ encompasses a much larger domain than $\mathbb{W}$.

Step 5: Final Conclusion

Because the set of rational numbers ($\mathbb{Q}$) contains fractions and negative integers—neither of which belong to the set of whole numbers ($\mathbb{W}$)—the proposition fails. The correct logical statement is that every whole number is a rational number, but not vice versa.

Final Solution: False. Rational numbers include fractions (e.g., $\frac{1}{2}$) and negative integers (e.g., $-3$), which are not whole numbers. Therefore, it is incorrect to state that every rational number is a whole number.